Absolute value of x = -x if x is negative explain?

Question

If the absolute value is the distance from zero then how can anything be a negative? Example: I-2I=2 I2I=2. Can someone explain why "absolute value of x = -x if x is negative"? This is the definition in the book.

Book Definition:
lxl=x if x is positive (or 0)
lxl=-x if x is negative

Answer 1

"absolute value of x = -x if x is negative"

looking at the line, you will notice that negative numbers are lined up on the left of 0, and positive numbers are on the right of 0. I.e. -5 would be five distances from 0 to the left, and 5 would be 5 distances from 0 to the right.
Therefore, while the absolute value tells you "how far", the signs + or - are denoting "where"/"on which side" the particular point is settled. And why the left side is considered negative - my guess is that "once upon a time...", right was considered right, correct, positive, etc., while left - as the opposite of right - was considered wrong, incorrect, negative. And it stayed that way. (It was only until recently - within last 50 years at the most - that left-handedness was recognized as OK and equal as right-handedness.)

Answer 2

Are you sure you read the book right?... cuz there is no way for |x| to be a negative number unless the negative is on the outside of the absolute value thingys.

Answer 3

I can give you some examples of - x

Suppose you have $5 in your checking account $5 = |5|
but you write a check for $10.
You are then $5 overdrawn. -$5 = |5|


Suppose you are 30 miles from Las Vegas 30 = |30|
and you drive into Las Vegas and go 30 miles past it. -30 = |30|

Answer 4

You're right that the distance from zero can't be negative.

What you're forgetting is that if x is negative, then -x will be positive.

Let me show an example:
|x| = x if x is positive
|2| = 2

|x| = -x if x is negatvie
|-2| = -(-2) = 2

Answer 5

a double negative is a positive

- (-1) = +1

so the absolute value of all numbers is positive. Basically, it is how far from zero a number is.

absolute number of positive numbers is the same number

absolute number of negative numbers is - (x)

Answer 6

The definition of absolute value, for x any real number,
is |x|=max{x, - x}
When x<=0, - x>=0;
Then - x>=x;
|x| = max{x, - x} = - x;
Proven.

Answer 7

If you read your addtional details closely, you will notice that the abosute value is always positive. This is what is meant by abosulute in this case.